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Advances in the Planarization Method:Effective Multiple Edge Journal of Graph Algorithms and Applications http://jgaa.info/ vol. 0, no. 0, pp. 0{0 (2012) Advances in the Planarization Method: Effective Multiple Edge Insertions Markus Chimani 1 Carsten Gutwenger 2 1Institute of Computer Science, FSU Jena 2Department of Computer Science, TU Dortmund Abstract The planarization method is the strongest known method to heuristi- cally find good solutions to the general crossing number problem in graphs: Starting from a planar subgraph, one iteratively inserts edges, represent- ing crossings via dummy vertices. In the recent years, several improvements both from the practical and the theoretical point of view have been made. We review these advances and conduct an extensive study of the algorithms' practical implications. Thereby, we also present the first implementation of an approximation algorithm for the crossing number problem of general graphs. We compare the obtained results with known exact crossing number solutions and show that modern techniques allow surprisingly tight results in practice. Submitted: Reviewed: Revised: Accepted: Final: Published: Article type: Communicated by: Regular Paper The research was partially supported by EUROGIGA project GraDR 10-EuroGIGA-OP-003. Markus Chimani was funded by a Carl-Zeiss-Foundation juniorprofessorship. E-mail addresses: [email protected] (Markus Chimani) [email protected] dortmund.de (Carsten Gutwenger) JGAA, 0(0) 0{0 (2012) 1 1 Introduction Given a graph G = (V; E), the crossing number problem asks how to draw G into the plane with the fewest possible number of edge crossings. The planarization method is the probably best known and most successful heuristic to tackle the crossing number problem in practice. In its simplest form it runs in two phases: first, a (large) planar subgraph G0 = (V; E0) of G is computed. Then, the temporarily removed edges F := E n E0 are re-inserted one after another, each time solving a single edge insertion problem. This problem can be stated as follows: Let H be a planar graph and e an edge not yet in H. We search for a smallest planar graph H+ which represents a drawing of H + e where edge crossings are replaced by dummy vertices of degree 4, and all these crossings occur on the edge e. Hence, when removing the image of e from H+, we obtain a planar embedded graph H. Using this method, each edge of F is inserted in a planar graph until we obtain a planarization G+, representing G in a planar way by using dummy vertices for crossings. In the first proposal [2] of this heuristic, the insertion problem was considered with respect to a fixed embedding (cyclic order of the edges around their incident vertices) of the planar graph H. (I.e., after obtaining the planar subgraph G0, one embedding of G0 is fixed and retained throughout the whole insertion phase.) A simple linear-time BFS-algorithm in the dual graph of H suffices to find an optimal solution. Later, and rather surprisingly, it was shown in [18] that there exists a linear-time algorithm, using the SPQR-tree data structure, which finds the optimal insertion path for e over all possible planar embeddings of H. In [17] it was shown that this approach is in practice vastly superior to the former in terms of the overall obtained number of crossings. In recent years it was furthermore shown that there exists a (rather com- plex) insertion algorithm [6] to optimally insert a vertex with all its incident edges into a planar graph H, considering all possible embeddings of H (ver- tex insertion). On the other hand, it is NP-hard to insert an arbitrary set of edges simultaneously (multiple edge insertion), even when the embedding of the planar graph into which to insert is fixed [35]. Most interestingly, the single edge insertion problem (over all possible em- beddings of H) is known to approximate the crossing number of H + e within a factor of ∆=2, where ∆ is the graph's maximum (vertex-)degree [4, 19]. I.e., there exists a drawing of H +e in which no two edges of H cross and whose num- ber of crossings is at most ∆=2 times the crossing number of H + e. Similarly, the vertex insertion problem approximates the crossing number of the resulting graph [8]. In particular, the proof of the latter can be generalized to show that an optimal multiple edge insertion solution|with respect to an edge set F and over all possible embeddings of G0|approximates the crossing number of G0 +F within a factor only dependent on ∆ and jF j [8]. Hence, the question arose whether this multiple edge insertion problem can be efficiently approximated. After a rather complicated approach in [10], a sim- pler and at the same time approximation-wise stronger algorithm was presented only recently [7]. The algorithm reuses concepts of the SPQR-tree based single 2 M. Chimani, C. Gutwenger Advances in the Planarization Method edge insertion and seems simple enough to be implemented and used in practice. The latter paper also shows that the traditional iterative single edge insertion algorithm cannot be an approximation strategy for the crossing number of G. Contribution. In this paper we present recent advances of the planarization approach from a practical point of view. On the one hand, we show how to improve on the traditional approach of iteratively inserting single edges, via the use of strong postprocessing routines. On the other hand, we give the first im- plementation of a simultaneous multiple edge insertion algorithm|hence, this is also the first practical study of any crossing number approximation algorithm for arbitrary graphs. By considering graph classes of known crossing numbers (either from theory or from the application of the currently strongest branch- and-cut based exact crossing minimization algorithm [9]) we can deduce a prac- tically very good performance of these heuristics: they usually find optimum, or at least very-close-to-optimum, solutions. In the next section we recapitulate the central decomposition data structures used for the insertion algorithms. We then summarize the general strategies and central steps of these algorithms chiefly in Section 3. Based thereon, Section 4 describes improvements and implementation and engineering choices, allowing the algorithms to work well in practice. Finally, Section 5 contains the full discussion of our experimental evaluation, and Section 6 concludes the paper with some interesting open problems. 2 Preliminaries In order to present our algorithmic choices and modifications, we first have to briefly introduce two central decomposition structures, used in all algorithms dealing with the insertion problem over all possible embeddings of H. Let H be a connected graph. The BC-tree B = B(H) of H is a tree with two different node types B and C: For each cut vertex in H, B contains a unique corresponding C-node, and for each block, i.e., a maximal two-connected subgraph or a bridge, in H a unique corresponding B-node. Two nodes in B are adjacent if and only if they correspond to a block b and a cut vertex c, such that c 2 b. It is well-known that the size of B is linear in the size of H, and that B can be constructed in linear time, by computing the biconnected components of H; see [21, 34]. Based thereon, we can further decompose each non-trivial block H0 (i.e., a block that is not a bridge) via an SPQR-tree [12, 13] into its triconnected components: While SPQR-trees are more complicated than BC-trees, they also only require linear size and can be constructed in linear time [16,20]. This data structure is particularly interesting, as it directly encodes all (exponentially many) planar embeddings of H0. We use the definition from [5, 7] which does not use Q-nodes, and therefore call the decomposition SPR-tree for conciseness. Chiefly summarizing, each tree node corresponds to a skeleton, a \sketch" of H0 where certain subgraphs are replaced by virtual edges; a skeleton's structure is JGAA, 0(0) 0{0 (2012) 3 Figure 1: A planar graph (top left), its SPR-tree (top right), and its decomposition showing the skeletons (bottom). Virtual edges are represented by dotted lines. restricted to only three simple types. By repeatedly merging the skeletons of adjacent nodes (at their virtual edges representing each other), we can obtain the original graph. See Figure 1 for an example. Definition 1 (SPR-tree). Let H0 be a biconnected graph with at least three vertices. The SPR-tree T = T(H0) of H0 is the (unique) smallest tree satisfying the following properties: i. Each node ν in T corresponds to a skeleton Sν = (Vν ;Eν ) which is a \sketch" (minor, in fact) of H: Certain subgraphs are replaced by single virtual edges. The non-virtual edges are referred to as original edges. ii. The tree has three different node types with specific skeleton structure: S: The skeleton is a simple cycle; it represents a serial component. P: The skeleton consists of two vertices and multiple edges between them; it represents a parallel component. R: The skeleton is a simple triconnected graph. Note that a planar tri- connected graph has a unique embedding (up to mirroring). iii. For the edge (ν; µ) in T we have: Sν contains a virtual edge eµ which represents the subgraph described by Sµ, and vice versa. 4 M. Chimani, C. Gutwenger Advances in the Planarization Method iv. We can obtain the original graph H0 by iteratively merging the skeletons of adjacent tree nodes: For the edge (ν; µ) in T, let eµ (eν ) be the virtual edge in ν (µ) representing the subgraph described by Sµ (Sν , respectively).
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